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Aug
31
answered If an unary language exists in NPC then P=NP
Aug
22
comment If $X_n$ are symmetric around $0$ then show $P(|S_n|\geq\max_{1\leq i\leq n}|X_i|)\geq\dfrac{1}{2}$
It's a rather standard trick that arises when dealing with symmetric variables. Generally speaking, after you've read many proofs, you will know most of the standard trick, and will be able to use them when necessary.
Aug
22
comment If $X_n$ are symmetric around $0$ then show $P(|S_n|\geq\max_{1\leq i\leq n}|X_i|)\geq\dfrac{1}{2}$
The trick here is to use some symmetry.
Aug
22
answered If $X_n$ are symmetric around $0$ then show $P(|S_n|\geq\max_{1\leq i\leq n}|X_i|)\geq\dfrac{1}{2}$
Aug
22
comment Let $A_n=\frac{(n+1)+(n+2)+(n+3)+…+(n+n)}{n}$,$B_n=[(n+1)(n+2)(n+3)…(n+n)]^{1/n}$.If $\lim_{n\to\infty}\frac{A_n}{B_n}=\frac{ae}{b}$
For me the Riemann sum argument is more complicated, since it requires a bit of thought, while using Stirling's approximation is a thoughtless calculation. To each their own.
Aug
22
answered Let $A_n=\frac{(n+1)+(n+2)+(n+3)+…+(n+n)}{n}$,$B_n=[(n+1)(n+2)(n+3)…(n+n)]^{1/n}$.If $\lim_{n\to\infty}\frac{A_n}{B_n}=\frac{ae}{b}$
Aug
22
comment Probability of hitting a number $Ib$ (rare case)
I give up. You should try harder solving your problems on your own, without constantly asking for outside help.
Aug
21
awarded  Nice Answer
Aug
17
awarded  Yearling
Aug
17
comment Probability of hitting a number $Ib$ (rare case)
@Turbo No, this formula is incorrect. Try again. Don't guess, calculate.
Aug
16
comment Probability of hitting a number $Ib$ (rare case)
Each of the $N^{1-\epsilon}$ numbers hits $T$ with probability $1/N$. Since I assumed that sampling is with replacement, the probability that none of them hit $T$ is $(1-1/N)^{N^{1-\epsilon}}$.
Aug
15
comment Probability of hitting a number $Ib$ (rare case)
The success probability in Chernoff's bound is often though of as constant, i.e., bounded away from 0 and 1. In this case it is very close to 0 or 1 (depending on what you count as success). Perhaps a more refined version of the bound would work. In any case, Chernoff's bound is only an upper bound, and is not guaranteed to give a reasonable bound in all cases.
Aug
15
comment Probability of hitting a number $Ib$ (rare case)
You probably didn't use enough terms in Stirling's approximation. For $\epsilon > 0$ it shouldn't matter too much whether you're choosing the elements with or without replacement, since even if you choose them with replacement, with high probability the number of unique elements is $\Theta(N^{1-\epsilon})$.
Aug
15
answered Probability of hitting a number $Ib$ (rare case)
Aug
15
comment Probability of hitting a number
I don't have an opinion on the matter, though usually computer algebra systems are right. If you use Stirling's approximation, you might be able to compute the constant.
Aug
15
comment Probability of hitting a number
That's an excellent opportunity to learn. Wikipedia has a good description.
Aug
15
comment Probability of hitting a number
Have you tried using Stirling's approximation here?
Aug
15
comment Probability of hitting a number
Then the behavior would be quite different. It would make a nice exercise for you.
Aug
13
comment Trace inequality on powers of non-negative matrix
@LaRias Using Jensen's inequality.
Aug
12
answered Trace inequality on powers of non-negative matrix